EMLAB

1

Uniform plane wave

EMLAB

2Radiation from a current filament

SR

ej

jkR

4)ˆ(ˆ)( JRRJrE

EMLAB

3Spherical waves from a source of finite dimension

R

eE

jkR

4

Transmitted waves from a finite sized source behave like spherical waves.

EMLAB

4Concept of plane waves

jkReE R

eCRE

jkR

4)(

)(

)()(

4)(4)(

RRjk

RRjkRRjk

eC

R

eC

RR

eCRRE

RR

R

)()( RRjkeCRRE

jkReCRE )(

EM waves transmitted from a finite sized source spread spherically in the space. At great distances from the source, EM waves behave as a plane wave locally.

EMLAB

5E & H in source free region

0

B

D

DJH

BE

t

t1. In source free region, E and H can be obtained easily.2. With J and ρ equal to zero, a source free wave equation is ob-

tained.

0

0

H

E

EH

HE

j

j02

22

EE

EEHE

k

kj

0

0)(22

222

EE

EEEEE

k

kk

22 k

EMLAB

6

022 EE k

• If the source current away from the field point has the direction of x-axis, the electric field can have only x component. If the extent of the source is infinite along x and y directions, variations of the field along those directions become zero.

022

2

xx Ek

dz

Ed

jkzx

jkzxx eEeEE 00

The first term represent the wave propagating toward +z direction, while the second stands for the wave propagating to the opposite direction. Considering only the wave propagating in the positive direction,

jkzx

x

jkzx

eEz

E

jj

eE

0

0

1ˆ

1ˆ

ˆ

yy

EH

xE

3770 wave impedance of free

space

• If the propagating direction is in +z-axis, the Helmholtz equation be-comes a simple expression.

02

2

2

2

y

E

x

E xx

EMLAB

7Plane wave propagating general direction

0

0

22

2

22

xx Ek

dz

Ed

k EE 1. If the propagating direction is other than x, y or z direction, the phase term in the exponential function can be obtained by the inner product of the k-vector and the position vector.

2. the k-vector can be compared to the normal vector of equi-phase plane.jkz

x eE 0xE

rkrk ExE ˆ

0

ˆ

0ˆ jkjkx eeE

)ˆˆˆˆ( zyxkk zyx kkkk

rkxE jx eE 0ˆ

)point nobservatio: , npropagatio waveof direction:ˆ( rk

EMLAB

8

)ˆˆˆˆ( zyxkk zyx kkkk rkxE jx eE 0ˆ

EkEkE

H

ˆˆ

j

jk

j

The magnetic field can be obtained.

EkEk

EEErk

rkrkrk

jej

eeej

jjj

0

000 )()(

Magnetic field of a plane wave

AAA uuu )(

Vector identity

EMLAB

9Wave propagation in dielectrics

1. If the wave is propagating into a medium other than free space, the molecules in that medium vibrate under the action of electric field. Meanwhile, the energy of the wave is dissi-pated and the medium become heated.

2. The dissipation can be modeled by a complex permittivity.

EE

EE

JJEEEJH

jjjj

jjjj

jj displcond

jj

3. The ratio of the real part of εr to the imaginary part is called the loss tangent of that media.

tan

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10

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11

4. From the equations on the left, it can be seen that the phase of displacement current leads that of conduction current by 90 degree.

5. That is, electric field propagates first, then charges move un-der the action of that electric field.

6. Helmholtz equation in a lossy medium becomes,

displcond JJEEEJH jj

EJ

EJ

jdispl

cond

0)(

0)(

22

2

22

xx Ej

dz

Ed

j

EE

j

jeEE zxx

)(, 220

Propagation constant in a lossy dielectric

EMLAB

12

222222

2222

22

)2

)(2)1

)(,

jj

jj

4

)/(12

22

2222

동순복호 1)/(12

1)/(12

2

)()/(1

04

)/(1

0)(

2

22

2/1224222422

2

2

2222

22222

tt

tt

zjzxx eEE 0

To obtain the approximate expression of α and β, we consider the two extreme cases of

① Good dielectric : ② Good conductor :

EMLAB

13

동순복호 1)/(12

2/12

Case 1)

22/1

2

2/12

22

)/(8

112

/2

1

2)/(4

112

/2

1

2

1)/(2

11

2

)/(2

11)/(1

동순복호

22

8

11)/(

8

11

22

Plane wave in good dielectrics

EMLAB

14

-5 -4 -3 -2 -1 0 1 2 3 4 5-1

-0.5

0

0.5

1

Plane wave in good conductors

동순 복호 1)/(12

2/12

depth)skin ;(1

2

동순 복호 /2

)/()/(1

2/1

2

zj

zzjz ee

00 EEE

δ is called as the skin depth of the medium at the given frequency. If the electric field pene-trate into a lossy medium as much as one skin depth, its strength decreases by 1/e. That is its strength becomes 36.7% of the original value.

conductor

depthskin :

Case 2)

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15

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16

EMLAB

17Example

Find the skin depth of sea water at the frequency of 1MHz. In sea water

jjj

1 1890

)1085.8)(81)(102(

4126

good conductor

]m[25.02

]m[6.12

depth) skin;(1

2

.81,]/[4 rmS

EMLAB

18Example

Calculate the resistance of a round copper of 1mm radius and 1km length at DC and 1MHz.

48.5)108.5(10

1076

3

2 r

lRDC

5.41)108.5)(10066.0(102

10

2 733

3

MHz1 a

lR

EMLAB

19Wave polarization

022 EE k 1. The Helmholtz equation has three components (x, y, z).2. But the divergenceless condition imposes that ∇∙E = 0,

which is a constraint among the vector components of E-field. That is, all the component of Ex, Ey, Ez are not independent.

00

z

E

z

E

x

E zyxE

0

zzyyxxzyx EkEjkEjkz

E

z

E

x

EE

From the above condition, only two components among the Ex, Ey, Ez are independent. That is, two kinds of independent po-larizations comprise an arbitrary E-field.

a changing direction of electric field observed at a position.

Among the components of an electric field vector, only two of them is independent.

EMLAB

20PolarizationExample of a electric polarization of a wave propagating in +z direction.

x

y Linear polarization

Circular polarization

EMLAB

21Polarization diversity antenna